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A352427
a(n) is the number of trailing 0's in the minimal representation of n in terms of the positive Pell numbers (A317204).
1
0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 4, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 4, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 5, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1
OFFSET
1,5
COMMENTS
The asymptotic density of the occurrences of 0 is sqrt(2)-1 and of the occurrences of k = 1, 2, ... is 2*(sqrt(2)-1)^(k+1).
The asymptotic mean of this sequence is 1 and its asymptotic variance is sqrt(2).
LINKS
FORMULA
a(A000129(n)) = n-1 for n>=1.
a(n) = 0 if and only if n is in A286666.
a(n) > 0 if and only if n is in A286667.
a(n) == 0 (mod 2) if and only if n is in A003152.
a(n) == 1 (mod 2) if and only if n is in A003151.
MATHEMATICA
pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerExponent[Total[3^(s - 1)], 3]]; Array[a, 100]
CROSSREFS
Similar sequences: A003849 (dual Zeckendorf), A035614 (Zeckendorf), A230403 (factorial), A276084 (primorial), A278045 (tribonacci).
Sequence in context: A190487 A054528 A025884 * A257024 A334996 A124433
KEYWORD
nonn,base
AUTHOR
Amiram Eldar, Mar 16 2022
STATUS
approved